Research Policy and U.S. Economic Growth

Transcription

1 Research Policy and U.S. Economic Growth Richard M. H. Suen August 4, 203. Abstract This paper examines quantitatively the e ects of R&D subsidy and government- nanced basic research on U.S. economic growth and consumer welfare. To achieve this, we develop an endogenous growth model which takes into account both public and private research investment, and the di erences between basic and non-basic research. A calibrated version of the model is able to replicate some important features of the U.S. economy over the period Our model suggests that government spending on basic research is an e ective policy instrument to promote economic growth. Subsidizing private R&D, on the other hand, has no e ect on economic growth. Keywords: Research Policy, Basic and Applied Research, R&D Spending, Endogenous Growth JEL classi cation: O3, O38, O4. Department of Economics, 365 Fair eld Way, Unit 063, University of Connecticut, Storrs CT richard.suen@uconn.edu. Phone: (860) Fax: (860)

2 Introduction A central implication of modern growth theory is that long-term economic growth is mainly driven by technological improvements. While certain improvements can be achieved through learning-by-doing (Arrow, 962; Romer, 986), most of the innovations we see today are the products of purposeful research and development. Over the past several decades, the U.S. government has adopted various policies to spur R&D investment and economic growth. In this study, we focus on two types of research policies, namely tax incentives for private R&D and government spending on basic research. The goal is to evaluate the e ects of these policies on U.S. economic growth and consumer welfare. To achieve this, we develop an endogenous growth model with both public and private research investment. A calibrated version of the model is able to replicate some important features of the U.S. economy over the period We then perform a series of counterfactual experiments to quantify the impact of research policies on economic growth and consumer welfare. The e ect of government policies on private R&D investment has long been a subject of interest among growth theorists. Since the pioneering work of Romer (990), Grossman and Helpman (99), and Aghion and Howitt (992), a substantial body of research has examined the determinants of private R&D investment and its relation to economic growth. 2 These so-called R&D-based growth models provide a micro-founded framework for analyzing research policies. Most of the existing studies, however, focus exclusively on the e ects of tax incentives (such as tax exemptions and tax credits) for private R&D and overlook the importance of direct government spending on innovative activities. 3 Since World War II, the U.S. government has played a crucial role in funding these activities. Although the overall importance of public R&D spending has declined since the 960s, the government remains the most important source of nancial support for one particular type of research, namely basic research. 4 Basic research is one of the three major categories of research activities, the other two being applied research and development. 5 Basic research refers to studies that are solely intended to advance our For more information about the U.S. national innovation system and other research policies, see Mowery (998) and Shapira and Youtie (200). 2 For a detailed review of these studies, see Aghion and Howitt (2005) and Jones (2005). 3 Examples include Grossman and Helpman (99), Jones (995), Peretto (998), Segerstrom (998, 2000), Young (998), Howitt (999), and Impullitti (200). Notable exceptions are Park (998), Morales (2004), and Akcigit et al. (202). The latter group of studies will be discussed in greater detail later on. 4 Between 953 and 2009, the share of total R&D spending paid by the U.S. government has dropped from 54.7 percent to 32.0 percent. Over the same time period, the government has paid more than 60 percent of all the expenses on basic research, and its importance has not declined. See Section 2 for a more detailed discussion on these trends. 5 This classi cation is adopted by the National Science Foundation (NSF) in the United States and the Organization for Economic Co-operation and Development (OECD). For a formal de nition of these research activities, see < fedgov.cfm> or the OECD Frascati Manual 2002, Section

3 knowledge on fundamental principles or fundamental aspects of phenomena (e.g., research on pure science and pure mathematics). In particular, this type of research is not directed towards any speci c application or commercial goal. Applied research and development, on the other hand, refer to studies that are targeted towards a speci c practical goal or the actual production of new products. Thus, the objectives of basic and non-basic research are fundamentally di erent. Empirical evidence shows that basic research often provides the basis for non-basic research (Mans eld, 995; Narin et al. 997), and it seems to have a larger contribution to productivity growth than non-basic research (Mans eld, 980; Griliches, 986). These two types of research also di er markedly in terms of funding source and performing sector. Basic research is mostly funded by the government and performed outside of the private sector, whereas applied research and development are mostly funded and performed by the private sector. For instance, 78.3 percent of all the expenses on basic research in 2009 were paid by the government (56.7 percent), universities (0.8 percent) and other non-pro t institutions (0.8 percent). The rest was paid by private industries. On the contrary, private industries paid 7. percent of all the expenses on applied research and development in 2009, while the government paid 26.2 percent. 6 In terms of performing sector, 80.5 percent of basic research was conducted by universities (53.3 percent), other non-pro t institutions (2.2 percent), federally funded research and development centers (7.7 percent), and federal agencies (7.2 percent). The rest was performed by private industries. This contrasts sharply with the fact that 82.0 percent of applied research and development was conducted in the private sector. 7 These observations suggest that it is important to distinguish between basic and non-basic research when analyzing the impact of public R&D spending. In this study, we develop an endogenous growth model which takes into account the major di erences between basic and non-basic research, and the importance of direct government spending on basic research. Our analysis also takes into account several empirical trends in the U.S. economy over the period , such as a rapid increase in the employment of researchers (relative to total employment), a persistent decline in corporate income tax, the introduction of R&D tax credit in 98, and other changes in scal policies. In our model framework, basic research is targeted towards the creation of fundamental knowledge, whereas non-basic research is directed at improving the quality of productive inputs. Fundamental knowledge is bene cial to society because it can improve rm productivity and enhance the 6 Most of the federal obligations for applied research and development are defense-oriented. For instance, the Department of Defense and the defense programs under the National Nuclear Security Administration accounted for 68.6 percent of these obligations in On the other hand, these federal agencies only accounted for 5.3 percent of all the federal obligations for basic research in the same year. 7 See Section 2 for more information about the patterns of R&D spending and how they change over time. 3

4 e ciency of non-basic research. This type of knowledge is freely available to all parties once it is discovered. Since there is no private ownership of fundamental knowledge, no rm is willing to invest in basic research. Hence, basic research is entirely funded by the government in our framework. 8 On the other hand, innovations created by non-basic research are protected by patents, which establish perpetual monopoly rights over the sale of the improved products. The pro t stream generated by these rights provides the sole incentive for conducting non-basic research. In this study, we con ne our attention to non-basic research that is funded and conducted by pro t-maximizing rms. 9 Both basic and non-basic research require the use of highly trained and specialized workers, or researchers. The supply of these workers is exogenously given and inelastic at any given point of time. The assumption of inelastic supply captures the fact that it requires a considerable amount of time to train someone to do research, hence the total number of researchers cannot respond immediately to changes in market conditions. In each period, an exogenous fraction of researchers is recruited by the government to conduct basic research. This in turn determines the scale of public R&D spending and the growth rate of fundamental knowledge. The accumulation of fundamental knowledge brought by basic research, and the continuous improvement of input quality made possible by non-basic research are the two driving forces of technological advancement in our model. Similar to the neoclassical growth paradigm, the model developed here displays a rich set of transitional dynamics which is jointly determined by capital accumulation and technological improvements. In the theoretical analysis, we show that the model economy will eventually converge to a unique long-run balanced-growth equilibrium that does not exhibit the scale e ect. 0 In terms of policy implications, our model suggests that government- nanced basic research is a strong impetus to long-term economic growth. On the contrary, subsidizing private R&D investment (either in the form of tax exemptions or tax credits) has no e ect on long-term economic growth. To further explore the growth and welfare implications of these research policies, we construct a parameterized version of the model and solve for the equilibrium time paths of all major economic variables using numerical methods. Under the baseline parameterization, our model is able to replicate the patterns of R&D investment rate and the 8 A detailed discussion on this assumption is provided in Section 2. 9 As mentioned in footnote 6, most of the government spending on non-basic research is defense-oriented. As pointed out by Mowery (200), the unusual institutional setting of defense R&D programs and the speci c nature of military inventions make it di cult to evaluate the impact of defense R&D on the entire economy. Consequently, much of our understanding on the e ects of defense R&D is derived from case studies on particular industries. While these studies have raised some interesting issues (such as the e ects of public procurement on private R&D, and the civilian innovations that are inspired by defense R&D), we do not pursue them here, but rather focus on the e ects of government basic research spending. 0 More precisely, our model predicts that long-term economic growth is independent of the size of population and the total number of researchers. Similar to the models of Jones (995), Segerstrom (998), Peretto (998), Howitt (999) and Jones and Williams (2000), our model predicts that long-term economic growth depends positively on the long-term population growth rate. However, perpetual growth in per-capita variables is still possible even if the size of population or the total number of researchers ceases to grow in the long run. 4

5 increase in real per-worker GDP in the United States over the period We then perform a series of counterfactual experiments and welfare analyses in order to gauge the e ects of research policies. There are four major ndings from the numerical analyses. First, subsidizing private R&D investment has no e ect on technological progress and economic growth in both the short run and the long run. This is because R&D subsidies can only stimulate the demand for innovative activities and the demand for researchers, but have no e ect on the supply of researchers which is inelastic at any point of time. Hence, subsidizing private R&D will only drive up the equilibrium wage rate for researchers without a ecting the equilibrium quantity of labor input in innovative activities. Consequently, it has no e ect on technological progress and economic growth. 2 This result is consistent with the empirical ndings reported by Goolsbee (998), and the ideas discussed in Romer (200). Our second major nding is that the rapid increase in the number of researchers (relative to total employment) is an important contributing factor to U.S. economic growth. Under the baseline parameterization, our model suggests that 2.3 percent of TFP growth and 25.8 percent of the growth in real per-worker output over the period can be attributed to this factor. Third, unlike R&D subsidies, direct government spending on basic research can e ectively promote technological progress and economic growth. Our model suggests that 2.5 percent of TFP growth and 4.3 percent of the growth in real per-worker output between 953 and 2009 can be attributed to the rising share of government basic research spending in total R&D expenditures. 3 Our results also suggest that about two-thirds of the long-term economic growth in the United States can be attributed to the accumulation of fundamental knowledge. Because of these positive growth e ects, a permanent increase in government basic research spending can induce signi cant welfare gains for consumers. Finally, our results point to the signi cance of general equilibrium e ects (or price e ects) when evaluating the welfare implications of research policies. As we mentioned earlier, the e ects of R&D subsidies are manifested solely on the equilibrium wage rate for researchers. As for government- nanced basic research, a permanent increase in this type of spending not only accelerates the pace of technological progress, it also drives up the equilibrium real interest rates. From an innovating rm s perspective, higher interest rates mean that the pro t stream generated by an improved product is now discounted at higher rates. This lowers the rm s incentives to invest in non-basic research and their demand for researchers. Thus, Throughout this paper, R&D investment rate is de ned as the share of total R&D spending in GDP. 2 There are two reasons why this result does not appear in other studies, such as Peretto (998), Segerstrom (998) and Impullitti (200). First, these studies do not distinguish between researchers and non-researchers. Instead, they assume that all workers are identical and freely mobile between research sector and production sector. Hence, the supply of labor for research activities is not inelastic. Second, these studies typically use labor as the numeraire which means wage rate is normalized to one in every period. Thus, R&D subsidies has no e ect on the equilibrium wage rate in their models. 3 According to our measure, the share of government basic research spending in total R&D expenditures has increased from 7.25 percent in 953 to 3.86 percent in The time-series data of this share is shown in Figure 8. 5

6 following a permanent increase in government basic research spending, the relative wage of researchers to non-researchers declines initially before rising to the new (and higher) long-run level. Ignoring the price e ects during the transition period will thus lead to an overestimation of the welfare gains from government- nanced basic research. Other studies that have explored the growth e ects of public R&D spending include Park (998), Morales (2004), and Akcigit et al. (202). The rst two studies have generalized the model of Romer (990) to allow for government- nanced research. Both of them nd that government R&D policies can have a signi cant impact on long-term economic growth. These studies, however, do not explore the quantitative implications of their models. The present study is closer in spirit to Akcigit et al. (202) in the sense that both have developed an endogenous growth model with basic and non-basic research, and explored the quantitative implications of the model. However, the theoretical constructs and quantitative analyses in these two studies are very di erent. In the model of Akcigit et al. (202), private rms can choose to operate in multiple industries and invest in both basic and applied research. Workers are assumed to be homogeneous and freely mobile across industries and across sectors. In their theoretical analysis, these authors focus on characterizing the variations of R&D spending across di erent types of rms in the long-run stationary equilibrium, and their implications to long-term economic growth. In the quantitative analysis, they estimate the long-run equilibrium solution of their model using rm-level data for France over the period In the current study, we focus on the temporal patterns of public and private research spending, and analyze the impact of research policies in both the short run and the long run. In the quantitative analysis, we calibrate our model using existing empirical estimates and aggregate data for the United States over the past several decades, and quantify the e ects of research policies that are speci c to the United States. The rest of this paper is organized as follows. Section 2 summarizes the empirical trends of R&D spending and the total number of researchers in the United States. Section 3 describes the model economy. Section 4 de nes the equilibrium of the model and presents the main theoretical results. Section 5 explains the choices of the baseline parameter values. Section 6 presents the numerical results. This is followed by some concluding remarks in Section 7. 6

7 2 Empirical Trends In this section, we brie y summarize the key patterns of R&D spending in the United States. 4 Figure shows the share of total R&D spending in U.S. GDP over the period Between 953 and 964, R&D investment rate increased rapidly from.36 percent to 2.88 percent. Since then, it has been maintained between two and three percent. Figure 2 shows the distribution of total R&D spending among basic research, applied research and development. Over the entire sample period, development accounted for more than 60 percent of total R&D spending in the United States, while applied research accounted for another 20 percent. Despite the importance of non-basic research, the share of basic research spending has been persistently increasing over the years. In 953, basic research accounted for 8.9 percent of total R&D spending. This increased to 9.0 percent in Next, we turn to the importance of government and private industries in funding and performing these research activities. Figure 3 shows the breakdown of basic research spending by funding source. Table shows the distribution of basic research output (measured in terms of scienti c publications) by performing sector. Throughout the entire sample period, the U.S. government played a predominant role in funding basic research. In particular, the government has funded about 70 percent of all the basic research performed in academia, which is the primary contributor of scienti c publications. Another important observation from Figure 3 and Table is that, while private industries have funded and performed a substantial amount of basic research, they have produced only a small fraction of the output. This seems to suggest that the amount private rms spend on basic research may not truly re ect their importance in the creation of fundamental knowledge. In a well-cited article on industrial R&D, Rosenberg (990) suggests two possible reasons why private rms would choose to invest in and maintain a basic research capability: First, this type of capability is crucial for planning and evaluating non-basic research. Second, it is also essential for understanding and exploiting the knowledge created by academic research. 5 Thus, according to this view, the main reason why private rms invest in basic research is to enhance their performance in non-basic research, not to create fundamental knowledge. While it is di cult to test this hypothesis rigorously, it does seem to explain the empirical evidence mentioned above. Given the relatively insigni cant role of private industries in the creation of fundamental knowledge, we choose to 4 Unless otherwise stated, all the data reported in this section were taken from National Patterns of R&D Resources: 2009 Data Update compiled by the NSF. 5 The same author also pointed out that for industrial R&D the distinction between basic and non-basic research is often not that clear. For instance, in the Survey of Industrial Research and Development conducted by the NSF, industrial basic research is de ned as the pursuit of new scienti c knowledge or understanding that does not have speci c immediate commercial objectives, although it may be in elds of present or potential commercial interest. This de nition can be found in Research and Development in Industry: , Appendix A. 7

8 abstract away from basic research funded by private rms. This assumption helps keep the dynamics of the model tractable and allows us to focus on the growth and welfare e ects of government basic research spending. Figure 4 shows the distribution of non-basic research spending by funding source. It is evident from this diagram that the importance of government spending on applied research and development has been declining since the 960s. Figure 5 shows that this decline is coincident with the cutbacks in defenserelated and space-related R&D programs. In the present study, we only take into account two types of research activities: (i) government- nanced basic research, and (ii) applied research and development performed by private industries. Thus, when comparing the model to data, we use the sum of R&D expenditures incurred by these activities as our measure of total R&D spending (see Appendix C for more details). For the period , our measure of total R&D spending represents 77.3 percent of the U.S. total. The implied R&D investment rates are shown in Figure (labelled as Our Measure ). Over the past several decades, the U.S. has also witnessed a rapid increase in the employment of researchers. Figure 6 shows the growth factor of the number of full-time R&D scientists and engineers employed by private industries and the growth factor of total employment in the United States. 6 Between 953 and 2009, the employment of private-sector researchers has increased by a factor of 8.75, while total U.S. employment has increased by a factor of The Model 3. Demographics The model economy is populated by two types of households, namely high-skilled (H) and low-skilled (L) households. The number of each type of household is constant over time and is normalized to one. Throughout this paper, an index s 2 fh; Lg is used to indicate household type. Each household comprises a growing number of in nitely-lived consumers. All consumers within the same household are identical. Let N s;t be the number of consumers in type-s household at time t: Let n s;t be the growth rate of type-s household between time t and t + ; so that N s;t+ = ( + n s;t ) N s;t ; for s 2 fh; Lg : The growth rates 6 Data on full-time R&D scientists and engineers employed in the private sector are collected by the Business Research and Development and Innovation Survey (BRDIS), conducted by the NSF. These data are available from < search_hist.cfm?indx=24>. Note that these data do not include academic researchers and researchers employed by the government. Data on the employment of these researchers are relatively scarce. See Footnote 43 and Appendix C for more information. Data on the number of total employed workers (over age 6) are obtained from the Bureau of Labor Statistics website. 8

9 are exogenously given and changing over time in a deterministic manner. 7 Total population at time t is given by N t = N H;t + N L;t ; with N 0 = : The share of high-skilled consumers in the population at time t is denoted by t N H;t =N t : 3.2 Commodities There are two types of commodities in this economy. First, there is a single nal good which can be used for consumption and investment. The price of nal good is normalized to one in every period. Second, there is a continuum of di erentiated intermediate goods which can only be used to produce nal goods. The set of intermediate goods is xed over time and is represented by [0; ] : At the beginning of time 0; all intermediate goods have the same initial quality which is normalized to one. The quality of these goods can be improved over time, but the occurrence of quality improvement is stochastic and idiosyncratic across products. Each improvement raises quality by a factor of > : Hence, after j 2 f; 2; :::g improvements, the quality of the improved product is j : The probability of quality improvement is determined by a number of factors, including R&D investment made by pro t-maximizing rms. The details of this will be explained later. Let J t (!) be the number of improvements realized before time t for intermediate good! 2 [0; ] : Then there are J t (!) + di erent quality grades of the same intermediate good available at time t: Throughout this section, we will use the pair (j;!) to indicate an intermediate good that is of type! and quality j : The price of intermediate good (j;!) at time t is denoted by p t (j;!) : Let Q t (!) be the quality level of product (J t (!) ;!) ; i.e., Q t (!) = {z ::: } : J t(!) times We will refer to this as the state-of-the-art quality for intermediate good!: The highest quality level among all intermediate goods at time t is represented by q t max fq t (!) :! 2 [0; ]g : We will refer to this as the leading-edge quality at time t: 3.3 Final-Good Sector In the nal-good sector, there is a continuum of identical rms which produce the same product. In each period, each nal-good producer hires low-skilled workers and intermediate inputs, and produces output 7 The terms high-skilled workers and researchers are used interchangeably throughout this paper. In the quantitative analysis, we use the time-varying growth rates of N H;t and N L;t to account for the empirical patterns in Figure 6. 9

10 according to Y t = t M t L Y;t ; with 2 (0; ) ; () where Y t denotes the quantity of nal goods produced at time t, L Y;t denotes the quantity of low-skilled labor employed, M t is a composite measure of intermediate inputs, and t is an economy-wide measure of total factor productivity (TFP). The composite measure M t is constructed as follows: Let M t (j;!) be the quantity of intermediate input (j;!) employed at time t: For each! 2 [0; ] ; de ne a quality-augmented measure of intermediate input, M f t (!) ; according to fm t (!) J t(!) X j=0 j M t (j;!) : (2) This measure is de ned as a weighted sum of all quality grades available for a given type of intermediate input. It has two main properties. First, high-quality inputs are weighted more heavily than low-quality ones, which means the former are more productive than the latter. Second, di erent quality grades of the same input are treated as perfect substitutes in the production of nal goods. These quality-augmented measures are then aggregated by a standard Dixit-Stiglitz aggregator to form the composite measure M t. Formally, this is given by Z M t 0 h i fmt (!) d! ; with 2 (0; ) : (3) The elasticity of substitution between any two types of intermediate inputs is given by = ( ) : 8 The productivity factor t is taken as exogenously given by the rms, and is determined by t X t ; with > 0; (4) where X t is the stock of fundamental knowledge created by basic research available at the beginning of time t. In the model economy, basic research is entirely funded by the government and its outcomes are freely accessible to all rms and consumers. In other words, fundamental knowledge is non-excludable. It is also non-rivalrous, which means the use of fundamental knowledge by one rm does not reduce its availability to others. Thus, all rms can bene t from the existing stock of fundamental knowledge to 8 Grossman and Helpman (99), Segerstrom (998), Dinopoulos and Segerstrom (999), and Impullitti (200) use the same quality-augmented measure and Dixit-Stiglitz aggregator to de ne a composite measure for a continuum of di erentiated products. The main di erence is that, in their models, the di erentiated products take the form of consumption goods rather than intermediate inputs in production. In addition, these studies focus on the special case in which = 0: 0

11 the same extent. Since fundamental knowledge is a public good, we will also refer to it as public R&D capital. Equation (4) states that public R&D capital is bene cial to rm productivity. This assumption is supported by the empirical ndings in Adams (990), Nadiri and Mamuneas (994), and Luintel and Khan (20). 9 An isoelastic function is used to ensure the existence of balanced-growth equilibria. The parameter can be interpreted as the elasticity of TFP with respect to public R&D capital. Since the production function in () is homogeneous of degree one, we can focus on the choices made by a single, price-taking rm. Let w L;t be the wage rate for low-skilled worker at time t: The representative rm s problem is given by max L Y;t ;M t() < Z J : X t(!) = tm t L Y;t w L;t L Y;t 4 p t (j;!) M t (j;!) 5 d! ; ; 0 j= subject to (2) and (3), and the non-negativity constraints on intermediate inputs, i.e., M t (j;!) 0 for all (j;!) : 20 The rst-order condition with respect to L Y;t is w L;t = ( Mt ) t : (5) L Y;t The rst-order condition with respect to M t (j;!) is " Mt fmt (!) p t (j;!) t L Y;t M t # j ; (6) which holds with equality if M t (j;!) > 0: The expression on the right-hand side of (6) is the marginal product of M t (j;!) : The rst-order condition in (6) has two main implications. First, if the price of input (j;!) is strictly greater than its marginal product, then the representative rm will choose to have M t (j;!) = 0: Second, for any given type of intermediate input, say! 2 [0; ] ; the rm will only purchase those quality grades that have the lowest quality-adjusted price, p t (j;!) = j : Let ep t (!) be the lowest quality-adjusted price for intermediate input! at time t. Following Segerstrom (998), we use the following assumption to break ties: If more 9 Using data over the period , Nadiri and Mamuneas (994) estimate the e ects of publicly nanced R&D capital on the productivity of twelve U.S. manufacturing industries. Their results suggest that public R&D capital has signi cant cost-saving e ects in these industries. Using U.S. data over the period , Adams (990) nd that knowledge created by academic research is an important contributor to productivity growth. Luintel and Khan (20) report similar results for a group of ten OECD countries over the period Given a Cobb-Douglas production function, it is never optimal for the rm to choose L Y;t = 0: Hence, we can ignore the non-negativity constraint on labor input. On the other hand, since di erent quality grades of the same intermediate input are treated as perfect substitutes in (2), the rm may choose to have M t (j;!) = 0 for some (j;!) : Thus, it is necessary to take into account the non-negativity constraints on intermediate inputs.

12 than one quality grades of the same type of intermediate input charge the same quality-adjusted price, then the representative rm will only purchase the one with the highest quality. This ensures that only one quality grade of each type of intermediate input will be used in every period. 2 If the representative rm chooses to use input (j;!) ; i.e., if M t (j;!) > 0; then its demand function is given by M t (j;!) = j P t [p t (j;!)] E t ; (7) where E t denotes total expenditure on intermediate inputs at time t; i.e., E t Z 2 3 JX t(!) 4 p t (j;!) M t (j;!) 5 d! = P t M t ; 0 j= and P t is a quality-adjusted price index de ned by Z P t [ep t (!)] d! : (8) 0 A formal derivation of these expressions can be found in Appendix A. 3.4 Intermediate-Goods Sector In the intermediate-goods sector, there is a continuum of in nitely-lived rms. Each rm is the original inventor of a single variety of intermediate good. 22 We assume that each original inventor has signi cant technical and cost advantages in improving its own product over any potential competitors so that it is the only researcher that attempts to improve its product. 23 Successful quality improvement is rewarded with a patent, which bestows perpetual monopoly right over the sale of the improved product. These assumptions imply the following market structure of the intermediate-goods sector. All rms in this sector are monopolistic in nature. Each monopolist specializes in the production and development of a single intermediate good. In particular, all available quality grades of the same product are supplied by a single monopolist. Each monopolistic rm is made up of two divisions: the manufacturing division and the R&D division. 2 This assumption also simpli es the pricing decisions faced by the suppliers of these inputs. See footnote 27 for more information. 22 The original invention process is not considered in here. 23 Barro and Sala-i-Martin (2004, Section 7.3) have formally shown that in a patent race between an incumbent rm and an outsider, if the former has signi cant cost advantages in quality improvement research over the latter, then it will be able to drive the outsider out of business and become the only researcher. We do not explicitly incorporate this type of patent race in our model to avoid further complicating the analysis. 2

13 The former is responsible for producing the product, and the latter is responsible for conducting quality improvement research Production and R&D Technologies All types of intermediate goods, regardless of quality and variety, can be produced by the same production technology which uses physical capital and low-skilled labor as inputs. This technology is given by Y M;t = t K t L M;t ; with 2 (0; ) ; (9) where Y M;t denotes the quantity of intermediate goods produced at time t, K t and L M;t denote the quantities of physical capital and low-skilled labor employed. The productivity factor t is the same as in (4). Similar to the nal-good producers, the monopolistic rms do not own the factors of production. Thus, they have to hire low-skilled workers and rent physical capital from the competitive factor markets in every period. As for the R&D process, we assume that high-skilled labor is the only private input employed in quality improvement research. 24 Let M;t be the quantity of high-skilled labor employed by an individual rm at time t; and let M;t be the average level of high-skilled labor input among all innovating rms in the intermediate-goods sector. The probability of progressing from quality q to q is given by t (q; M;t ) # M;t # M;t ; with # 2 (0; ) ; (0) t (q) where t (q) is a measure of R&D di culty at time t. This measure is taken as exogenously given by individual rms, and is determined by t (q) (q; q t ) X % t ; with % > 0: () The function (q; q t ) is assumed to be strictly positive, strictly increasing in the quality of the product (q) ; and strictly decreasing in the leading-edge quality (q t ) : 24 In the existing R&D-based growth models, it is typical to assume that labor is the only input employed in R&D activities. See for instance, Aghion and Howitt (992), Young (995), Segerstrom (998), Dinopoulos and Segerstrom (999), Peretto (999), Aghion et al. (200), Impullitti (200) and Chu et al. (202). Empirical evidence shows that labor compensation is the largest component in private R&D expenditures. According to Dougherty et al. (2007, Table 3), labor costs account for 46.5 percent of total manufacturing R&D expenditures in the United States in 997. The corresponding gures for France, Germany and Japan are 52.8 percent, 6.7 percent and 42.7 percent, respectively. Materials and supplies accounted for less than 20 percent of total manufacturing R&D spending in these countries. MacDonald (973, Appendix Table V) reports similar ndings for a group of ten OECD countries in

14 Equations (0) and () capture four fundamental aspects of quality improvement research. First, the probability of successful quality improvement is strictly increasing in the rm s own labor input. Since # 2 (0; ), this type of input is subject to diminishing marginal returns. The concavity assumption is consistent with the empirical ndings in Pakes and Griliches (984) and Hall et al. (986). Second, the di culty of quality improvement research is strictly decreasing in the existing stock of fundamental knowledge. In other words, fundamental knowledge is bene cial to the e ciency of private R&D. This assumption is supported by the empirical ndings in Ja e (989), Mans eld (995, 998), Anselin et al. (997) and Zucker et al. (998). Third, holding other things constant, high-quality products are more di cult to develop than low-quality ones. Thus, it becomes increasingly di cult to move up the quality ladder. This feature is captured by the assumption that (q; q t ) is strictly increasing in q for all q t : Finally, research carried out by one rm will also create external bene ts for other innovating rms. 25 This type of externality, or knowledge spillover, can occur in two ways: (i) when researchers communicate and interact with each other, and (ii) when individual researchers learn from other people s success (e.g., by reading the codi ed part of a patent or by reverse engineering). The expression # M;t in (0) is intended to capture the bene ts created by the rst channel. The main idea behind this expression is as follows: As the size of the research community expands, more interactions among researchers will be available and so the e ects of knowledge spillover will become larger. To ensure the existence of balanced-growth equilibria, we assume that the function in (0) is homogeneous of degree one in M;t and M;t : The second channel of knowledge spillover is captured by the assumption that (q; q t ) is strictly decreasing in q t for all q: In words, this means all innovating rms can learn and bene t from the leading-edge improvement, so that an increase in q t will increase their research e ciency. 26 The rm s research outlay at time t is given by w H;t M;t ; where w H;t is the wage rate for high-skilled labor. These expenses are subject to two types of preferential tax treatment which we will explain later. If an improved product is developed at time t, then the rm can start producing it as a monopoly at time t + : In each period, each monopolistic rm has to make two groups of decisions. First, it has to decide how much to produce and what price to charge for its product. Second, it has to decide how much to 25 There is now a large body of empirical studies which con rm the existence and signi cance of knowledge spillover across rms. See, for instance, Ja e (986), Audretsch and Feldman (996, 2004) and references therein. 26 The variable t () is similar in spirit to the R&D di culty index de ned in Segerstrom (998), Dinopoulos and Segerstrom (999) and Impullitti (200). In the rst study, the di culty index is determined by the past history of R&D investment in each industry, and is thus di erent across industries. In Impullitti (200), the di culty index is identical for all industries and directly proportional to the size of total population. Dinopoulos and Segerstrom (999) have considered the e ects of trade liberalization under both speci cations. These studies, however, do not consider the e ects of fundamental knowledge on the di culty of private R&D. 4

15 invest in quality improvement research. Since the monopolists do not own the factors of production, the choices of physical capital and low-skilled labor in the production stage are static in nature and do not interfere with the research investment decisions. Likewise, the choice of high-skilled labor input in the research stage has no direct impact on the production process. Thus, the two groups of decisions can be analyzed separately Production and Pricing Decisions Since all nal-good producers use only one quality grade of each type of intermediate input in any period, each intermediate-good producer will only supply one quality grade of its products. 27 In particular, these producers will always choose to produce the best quality grade of their products. To see this, suppose the supplier of intermediate good! 2 [0; ] chooses to produce quality grade j at time t; for some j 2 f0; ; :::; J t (!)g : Given a Cobb-Douglas production function, the minimum cost of producing M 0 units of this product is C t by M t M; where t is the unit cost of production. This cost is determined t Rt wl;t ; (2) t where R t is the rental price of physical capital at time t: Low-skilled workers are freely mobile across sectors, so there is only one wage rate (w L;t ) for these workers. Since the production technology is identical for all types of intermediate goods, so is the unit cost of production. The monopoly price of product (j;!) can be obtained by solving t (j;!) max p t(j;!) f[p t (j;!) t ] M t (j;!)g ; where the market demand function M t (j;!) is stated in (7). 28 The optimal monopoly price involves a proportional mark-up over the unit cost and is identical for all (j;!). Formally, this is given by p t (j;!) = p t t ; for all (j;!) : (3) 27 This avoids the intricate issue of how a multiproduct monopolist would set the prices of its products, an issue that is not directly related to the main objective of this study. For more discussion on this type of pricing problem, see Chu et al. (2008) and references therein. 28 Since there is a continuum of rms in the intermediate-goods sector, each rm is in nitesimal when compared to the entire sector. Thus, the aggregate price index P t is treated as exogenously given when solving the monopoly pricing problem. 5

16 The market demand function and the monopoly pro t function are then given by M t (j;!) = j P t t E t ; t (j;!) = ( ) j P t t E t : Note that monopoly pro ts are strictly increasing in product quality j. Thus, the monopolist will only produce the best quality grade J t (!) in any period t. Note also that both M t (J t (!) ;!) and t (J t (!) ;!) depend on the index! only indirectly through J t (!) : Thus, for an intermediate input with state-of-theart quality Q t (!) = q; the market demand function and the monopoly pro t function can be expressed as t M t (q) = (qp t ) E t ; (4) t (q) = ( ) qpt t E t : (5) From this point onward, we will drop the index! and identify each intermediate-good producer by the state-of-the-art quality of its product Research Investment Decisions Consider a monopolistic rm with state-of-the-art quality q at time t: The rm s pro ts t (q) are subject to corporate income tax. The tax rate at time t is denoted by p;t 2 (0; ). When deciding how much to invest in quality improvement research, the rm takes into account two types of preferential tax treatment on private R&D expenses. First, these expenses are fully deductible from corporate income tax. Second, these expenses are also eligible for additional subsidies, or R&D tax credits. The rate of subsidy at time t is d;t 2 (0; ) : Both the corporate income tax rate and the R&D subsidy rate are changing over time in a deterministic manner, and they are fully anticipated by the rm. In each period, the rm distributes its pro ts (net of taxes and research outlay) as dividends to its owners. Each monopolistic rm has a xed number of shares outstanding in the equity market. The total number of shares issued by each rm is constant over time and is normalized to one. Each of these shares is a claim to the rm s future dividends. The value of the rm is then given by the present discounted value of its future dividends. Formally, let 6

17 t (q) be the amount of dividend distributed at time t; which is de ned as t (q) ( p;t ) [ t (q) w H;t M;t ] + d;t w H;t M;t = ( p;t ) t (q) ( p;t d;t ) w H;t M;t : (6) Let V t (q) be the value of a rm with state-of-the-art quality q at time t: The value function is de ned recursively as V t (q) = max t (q) + t (q; M;t ) V t+ (q) + [ M;t + r t+ t (q; M;t )] V t+ (q) ; (7) subject to (0) and (6). The expected future value of the rm is discounted using the market discount rate ( + r t+ ) : Since # 2 (0; ), it is never optimal for the rm to choose M;t = 0. In other words, all intermediategood producers will invest in quality improvement research in every period. The rst-order condition for M;t is given by ( p;t d;t ) w H;t = # t (q) M;t M;t # Vt+ (q) V t+ (q) + r t+ : (8) Equation (8) states that optimality is achieved when the marginal cost of hiring high-skilled labor equals its marginal bene t. The marginal cost is the subsidized wage rate. The marginal bene t is determined by two factors: (i) the increase in success rate brought by an additional unit of high-skilled labor, and (ii) the present value of the net gain from the improved product. Note that a successful improvement from quality q to q not only creates some new value for the rm [i.e., V t+ (q)], but also destroys the continuation value of the original product [i.e., V t+ (q)] which now becomes obsolete. Hence, the bene t of successful research is measured by the net gain in the rm s future value. Holding other things constant, an increase in either the corporate income tax rate ( p;t ) or the R&D tax credit ( d;t ) will lower the marginal cost of hiring high-skilled labor and encourage the rm s demand for research input. 3.5 Distribution of Product Quality Since the timing and frequency of quality improvement is idiosyncratic across intermediate goods, a nondegenerate distribution of product quality emerges in every period t : The distribution of product quality is de ned over the support Q ; ; 2 ; ::: : Consider those intermediate goods with state-of- 7

18 the-art quality q 2 Q at the beginning of time t: The share of these goods is denoted by f t (q) 2 [0; ] ; with P q2q f t (q) = for all t: The initial distribution is degenerate and is denoted by f 0 () = : For future references, de ne an aggregate quality index Q t according to 8 < X Q t = f : t (q) q q2q 9 = ; : (9) The initial degenerate distribution then implies Q 0 = and q 0 = : The evolution of the quality distribution can be characterized as follows. Let M;t (q) be the optimal quantity of high-skilled labor employed by an intermediate-good producer with state-of-the-art quality q at time t: The probability of successful improvement is then given by b t (q) t [q; M;t (q)] : Starting from the initial condition f 0 () = ; the quality distribution changes over time according to f t+ (q) = b h i t (q) f t (q) + t b (q) f t (q) ; (20) for all q 2 ; ; 2 ; :::, and f t+ () = h i t b () f t () : (2) At time t + ; the set of intermediate goods with state-of-the-art quality q is the combination of two groups. First, among those with state-of-the-art quality q at time t; a fraction b t (q) of them will be upgraded to q at time t + : The size of this group is given by b t (q) f t (q) : Second, among those with h i state-of-the-art quality q at time t; a fraction t b (q) of them will fail to improve and remain at h i the same quality level. The size of this group is given by t b (q) f t (q) : Equation (20) states that f t+ (q) is given by the sum of these two groups. Equation (2) states that the set of intermediate goods with q = are those who fail to improve in every period. Note that, in every period t; there is always a strictly positive fraction of rms with leading-edge quality q t that succeed in improving their products. Thus, the leading-edge quality level will evolve according to q t+ = q t with probability one. 3.6 Basic Research In the model economy, a continuum of basic research projects is available in every period. Each project is indexed by a positive scalar z which indicates its quality. The set of research projects available at time t is uniformly distributed over the range [0; z t ] : The upper boundary z t represents the most advanced 8

19 project at time t: This value is assumed to be changing over time in a deterministic manner. The speci cs of this will be explained later. In each period, each member of the high-skilled household draws a single basic research project from the uniform distribution. The draws are independent over time and across individuals. After observing z; a high-skilled consumer then seeks outside funding in order to develop the project. If the project is unfunded, then the consumer will work as a researcher in the intermediate-goods sector (the private sector). Even if outside funding is available, the consumer can still decide whether or not to proceed with the project. If he chooses not to proceed, then again he will work in the private sector. If he chooses to proceed, then he will have to invest his own time endowment (which is normalized to one) on the project. Similar to quality improvement research, basic research only requires high-skilled labor as input and its success is uncertain. If a project of quality z is successfully developed at time t, then it will create (z) X t units of new fundamental knowledge. The function : [0; z t ]! R + is continuous and strictly increasing, which means high-quality projects contribute more to the growth rate of fundamental knowledge than low-quality ones. No new knowledge is created if the project fails. The probability of successfully developing a type-z project at time t is given by x;t (z) = x;t (z) : (22) In the above expression, the numerator represents individual researcher s own labor input (which is one), and the denominator indicates the di culty of developing a type-z project at time t: The di culty index is determined by x;t (z) x (z) = (Q t ) X % t ; with % > 0: (23) Both x : R +! R ++ and = : R +! R + are continuous, strictly increasing functions. Equation (23) captures three important aspects of basic research. First, the di culty of basic research in general is strictly decreasing in the available stock of fundamental knowledge. This captures the idea that past discoveries of basic research are bene cial to the productivity of current research. 29 Second, high-quality projects are more di cult to develop than low-quality ones. This assumption is captured by the strictly increasing function x () : Third, as the economy becomes more technologically advanced, basic research projects become more di cult for researchers to develop individually. This assumption 29 This mechanism is often referred to as the intertemporal knowledge spillover e ect. Note that X t enters into () and (23) in the same way. This assumption is needed to ensure the existence of balanced-growth equilibria. 9

20 is captured by the strictly increasing function = (Q t ) ; where Q t is used as a measure of technological development. The main ideas behind this assumption are as follows: A basic research project can be viewed as a collection of tasks. As technology advances, more sophisticated and specialized techniques for conducting basic research are available and adopted. As a result, the tasks involved in basic research projects also become more specialized. This makes it more di cult for a single researcher to complete all the tasks in these projects. This hypothesis provides a potential explanation for the increasing prevalence of collaboration and teamwork among scientists. 30 In particular, it is consistent with the fact that scienti c collaboration is more common in elds that require complex instrumentation (or big science ). 3 The function = (Q t ) represents one of two channels through which innovations brought by quality improvement research can feedback into basic research. The second channel is built upon the idea that major breakthroughs in non-basic research or technology often open up new areas of scienti c studies. 32 To capture this mechanism in the most parsimonious manner, we set z t = q t for all t: In words, this means a leading-edge improvement in technology will also broaden the range of scienti c research. 33 Under this speci cation, the upper bound z t will increase deterministically over time according to z t+ = z t : Next, we turn to the supply side of basic research funding. As we mentioned earlier, due to the publicgood nature of fundamental knowledge, no rm is willing to invest in basic research. Consequently, basic research projects are solely funded by the government. 34 The amount and allocation of basic research funding is determined by a national research agency. In each period, the government agency chooses a set of research projects so as to maximize the growth rate of fundamental knowledge subject to a policy constraint. Let D t [0; z t ] be a non-empty set of research projects funded by the government. For those research projects with quality z 2 D t ; a fraction x;t (z) of them will succeed and create (z) X t units of 30 For the empirical evidence on the rising trend of scienti c collaboration, see Beaver and Rosen (979) and Stephan (996, Table 4). Other possible explanations for this phenomenon have been examined by Katz and Martin (997) and Jones (2009). 3 For instance, Katz and Martin (997) and Klein (2005) point out that many projects in high-energy physics, astronomy, oceanography, and life sciences (e.g., the Human Genome Project) require sophisticated and specialized equipments such as particle accelerators, observatories, ocean research vessels, and supercomputers. Consequently, research in these areas typically require collaborations between specialists in di erent elds. In other words, it is very di cult (if not impossible) for a single researcher to perform all the tasks involved in these research projects. 32 See, for instance, Rosenberg (990, p.69) for some speci c examples. 33 Our theoretical analysis can be easily extended to accomodate a more general relationship between q t and z t: For instance, one can assume that z t is strictly increasing in q t and { t; where { t represents other exogenous factors that can also a ect the range of scienti c research. The assumption that q t and z t share the same exogenous growth rate is not essential for our major results. 34 This also explains why high-skilled workers will not self- nance any unfunded projects: Since there is no market for fundamental knowledge, the gain from a self- nanced project is zero, but the opportunity cost (w H;t) is always strictly positive. 20